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Large derivatives, backward contraction and invariant densities for interval maps
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Bruin, Henk, Rivera-Letelier, J., Shen, Weixiao and Strien, Sebastian van (2008) Large derivatives, backward contraction and invariant densities for interval maps. Inventiones Mathematicae, Vol.172 (No.3). pp. 509-533. doi:10.1007/s00222-007-0108-4 ISSN 0020-9910.
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Official URL: http://dx.doi.org/10.1007/s00222-007-0108-4
Abstract
In this paper, we study the dynamics of a smooth multimodal interval map f with non-flat critical points and all periodic points hyperbolic repelling. Assuming that vertical bar Df(n)(f(c))vertical bar -> infinity as n -> infinity holds for all critical points c, we show that f satisfies the so-called backward contracting property with an arbitrarily large constant, and that f has an invariant probability mu which is absolutely continuous with respect to Lebesgue measure and the density of mu belongs to L-p for all p < l(max)/(l(max) - 1), where l(max) denotes the maximal critical order of f. In the appendix, we prove that various growth conditions on the derivatives along the critical orbits imply stronger backward contraction.
| Item Type: | Journal Article | ||||
|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics | ||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
| Library of Congress Subject Headings (LCSH): | Derivatives (Mathematics) | ||||
| Journal or Publication Title: | Inventiones Mathematicae | ||||
| Publisher: | Springer | ||||
| ISSN: | 0020-9910 | ||||
| Official Date: | June 2008 | ||||
| Dates: |
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| Volume: | Vol.172 | ||||
| Number: | No.3 | ||||
| Number of Pages: | 25 | ||||
| Page Range: | pp. 509-533 | ||||
| DOI: | 10.1007/s00222-007-0108-4 | ||||
| Status: | Peer Reviewed | ||||
| Publication Status: | Published | ||||
| Access rights to Published version: | Restricted or Subscription Access | ||||
| Funder: | Engineering and Physical Sciences Research Council (EPSRC), Chile. Comisión Nacional de Investigación Científica y Tecnológica (CONICYT), Zhongguo ke xue yuan [Chinese Academy of Sciences] (CAS), 973 Program | ||||
| Grant number: | GR/S91147/01 (EPSRC), PBCT ADI-17 (CONICYT) |
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